Cognitive linguistics is an emerging discipline that studies the cognitive properties of natural languages and the cognitive models of languages in computing linguistics, cognitive computing, and computational intelligence. This paper presents the theoretical framework of cognitive linguistics in order to systematically formalize the syntaxes and grammars of natural languages. An abstract language model of cognitive linguistics is created at the top level. Based on it, the cognitive structures

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... f languages at the levels of lexis, phrase, clauses, sentence, paragraph, and article are formally modeled from the bottom up. Using contemporary denotational mathematics, the deductive grammar of English is formally modeled and rigorously analyzed. This basic research provides support for a wide range of applications in computational linguistics, cognitive informatics, online text processing, web search engines, machine language comprehension, autonomous machine learning, smart cell phones, semantic computing, and computing with words.